–6x + 10y = –6 This Is Solve A System Of Equations Using Substitution
Introduction
In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of equations using substitution, specifically the equation -6x + 10y = -6.
What is Substitution?
Substitution is a method of solving a system of equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is already solved for one variable. In this case, we will use the equation -6x + 10y = -6 and substitute the expression for x from this equation into the other equation.
Step 1: Solve One Equation for One Variable
To solve the equation -6x + 10y = -6 for x, we need to isolate x on one side of the equation. We can do this by adding 6x to both sides of the equation, which gives us:
-6x + 6x + 10y = -6 + 6x
Simplifying the equation, we get:
10y = 6x - 6
Now, we can solve for x by dividing both sides of the equation by 6:
x = (10y + 6) / 6
Step 2: Substitute the Expression for x into the Other Equation
Now that we have the expression for x, we can substitute it into the other equation. Let's say the other equation is 2x + 3y = 7. We can substitute the expression for x from the previous step into this equation:
2((10y + 6) / 6) + 3y = 7
Simplifying the equation, we get:
(20y + 12) / 6 + 3y = 7
Multiplying both sides of the equation by 6 to eliminate the fraction, we get:
20y + 12 + 18y = 42
Combining like terms, we get:
38y + 12 = 42
Subtracting 12 from both sides of the equation, we get:
38y = 30
Dividing both sides of the equation by 38, we get:
y = 30 / 38
Simplifying the fraction, we get:
y = 15 / 19
Step 3: Find the Value of x
Now that we have the value of y, we can substitute it back into the expression for x from the previous step:
x = (10y + 6) / 6
Substituting y = 15 / 19 into this equation, we get:
x = (10(15 / 19) + 6) / 6
Simplifying the equation, we get:
x = (150 / 19 + 6) / 6
Multiplying both sides of the equation by 6 to eliminate the fraction, we get:
x = (150 / 19 + 6 * 19 / 19) / 6
Simplifying the equation, we get:
x = (150 + 114) / 19
Combining like terms, we get:
x = 264 / 19
Simplifying the fraction, we get:
x = 264 / 19
Conclusion
In this article, we solved a system of equations using substitution, specifically the equation -6x + 10y = -6. We first solved one equation for one variable, then substituted the expression for that variable into the other equation. We then solved for the other variable and finally found the value of the first variable. This method is useful when one of the equations is already solved for one variable.
Example Problems
- Solve the system of equations using substitution:
2x + 3y = 7
x - 2y = -3
- Solve the system of equations using substitution:
x + 2y = 4
3x - 2y = 5
Tips and Tricks
- Make sure to check your work by plugging the values of the variables back into the original equations.
- Use a calculator to simplify fractions and decimals.
- Use a graphing calculator to graph the equations and find the intersection point.
References
- "Algebra and Trigonometry" by Michael Sullivan
- "College Algebra" by James Stewart
- "Mathematics for the Nonmathematician" by Morris Kline
Frequently Asked Questions (FAQs) About Solving Systems of Equations Using Substitution =====================================================================================
Q: What is substitution in solving systems of equations?
A: Substitution is a method of solving a system of equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is already solved for one variable.
Q: How do I know which equation to solve for first?
A: You can choose either equation to solve for first, but it's often easier to solve for the variable that appears in both equations. In the example we used earlier, we solved for x first because it appeared in both equations.
Q: What if I get stuck or make a mistake?
A: Don't worry! It's easy to get stuck or make a mistake when solving systems of equations. Take a step back, review your work, and try again. If you're still having trouble, you can try using a different method, such as graphing or elimination.
Q: Can I use substitution to solve systems of equations with more than two variables?
A: Yes, you can use substitution to solve systems of equations with more than two variables. However, it can be more complicated and may require more steps.
Q: How do I know if I've found the correct solution?
A: To check if you've found the correct solution, plug the values of the variables back into the original equations. If the equations are true, then you've found the correct solution.
Q: What if I get a fraction or decimal as my answer?
A: If you get a fraction or decimal as your answer, you can simplify it by dividing the numerator and denominator by their greatest common divisor (GCD) or by using a calculator.
Q: Can I use substitution to solve systems of equations with fractions or decimals?
A: Yes, you can use substitution to solve systems of equations with fractions or decimals. However, you may need to use a calculator to simplify the fractions or decimals.
Q: How do I know if I've used the correct method?
A: To check if you've used the correct method, review your work and make sure you've followed the steps correctly. If you're still unsure, you can try using a different method or asking a teacher or tutor for help.
Q: What if I'm still having trouble?
A: Don't worry! It's okay to ask for help if you're still having trouble. You can ask a teacher, tutor, or classmate for help, or you can try using online resources or video tutorials.
Common Mistakes to Avoid
- Not checking your work: Make sure to plug the values of the variables back into the original equations to check if they're true.
- Not simplifying fractions or decimals: Make sure to simplify fractions or decimals by dividing the numerator and denominator by their GCD or by using a calculator.
- Not using the correct method: Make sure to use the correct method for solving the system of equations.
- Not reviewing your work: Make sure to review your work and check for errors before submitting your answer.
Conclusion
Solving systems of equations using substitution can be a powerful tool for solving equations with multiple variables. By following the steps and avoiding common mistakes, you can find the correct solution and become more confident in your math skills. Remember to ask for help if you're still having trouble, and don't be afraid to try different methods or resources until you find one that works for you.